Palaeoclimatic information can be retrieved from the diffusion of the
stable water isotope signal during firnification of snow. The
diffusion length, a measure for the amount of diffusion a layer has
experienced, depends on the firn temperature and the accumulation
rate. We show that the estimation of the diffusion length using power
spectral densities (PSDs) of the record of a single isotope species can
be biased by uncertainties in spectral properties of the isotope signal prior
to diffusion. By using a second water isotope and calculating the
difference in diffusion lengths between the two isotopes, this problem
is circumvented. We study the PSD method applied to two isotopes in
detail and additionally present a new forward diffusion method for
retrieving the differential diffusion length based on the Pearson
correlation between the two isotope signals. The two methods are
discussed and extensively tested on synthetic data which are generated
in a Monte Carlo manner. We show that calibration of the PSD method
with this synthetic data is necessary to be able to objectively
determine the differential diffusion length. The correlation-based
method proves to be a good alternative for the PSD method as it yields
precision equal to or somewhat higher than the PSD method. The use of
synthetic data also allows us to estimate the accuracy and precision
of the two methods and to choose the best sampling strategy to obtain
past temperatures with the required precision. In addition to
application to synthetic data the two methods are tested on
stable-isotope records from the EPICA (European Project for
Ice Coring in Antarctica) ice core drilled in Dronning Maud Land,
Antarctica, showing that reliable firn temperatures can be
reconstructed with a typical uncertainty of 1.5 and
2

The stable water isotope ratio in precipitation is known to be related
to the local atmospheric temperature

Another method for extracting the temperature signal from the stable
water isotope signal is based on the diffusion of the isotope signal
in the firn stage. After snow falls onto the surface of an ice sheet,
its isotope signal is subject to diffusional smoothing

One of the main drawbacks of the PSD technique is that it
is dependent on parameter choices. In

To circumvent the issues related to the PSD method, we propose an
alternative method for finding the differential diffusion length based
on the correlation between the

In the firn stage the isotopic profile changes with time due to the
diffusion process and by thinning of the layers due to densification
and ice flow. The change in the isotopic concentration can be expressed
as

Inspection of the terms in Eq. (

The diffused isotope signal at any time

This illustrates that together with an independently derived
accumulation rate and an estimate of the thinning function
a temperature history can be reconstructed by determining the firn
diffusion length in the ice. This can in principle be done for either
of the water isotopes, but it is better constrained when the combination
of two isotopes is used, as will be shown in Sect.

To be able to verify the accuracy and precision of the reconstructed
values for the differential diffusion length, it is necessary to know
the true values. As this is impossible for data from existing ice core
records, we created synthetic data sets for this purpose. To give
statistically robust results, these data sets were created in a Monte
Carlo (MC) routine in which several parameters characterising the
resulting data set were randomised as is outlined below. In total 4000
data sets were created for each MC run. Within an MC run the
parameters that determine the diffusion length (firn temperature, mean
accumulation rate and thinning of the ice) were the same for each data
set, but parameters such as the amount of precipitation per event and
the amplitude of the isotope signal were varied. In this paper six
different runs will be discussed. The first MC run assumes climate
conditions similar to present-day climate at the European Project for
Ice Coring in Antarctica (EPICA) drill site in Dronning Maud Land
(EDML) and no vertical strain (thinning

The creation of a synthetic data set consists of several steps. First
a precipitation record for both

Climatological and glaciological parameters used in the different Monte Carlo runs. The firn temperature, annual accumulation and thinning are input values, leading to the (differential) diffusion lengths given in the last three columns.

In creating the precipitation signals it is assumed that on average
the isotope concentration of precipitation water varies sinusoidally
in time (with a period of 1

The effects of diffusion and densification on the isotope signals are
calculated numerically in discrete time steps. A finite-difference
technique is used to calculate the effect of diffusion for each time
step, with the firn diffusivity set by Eq. (

Finally, from each data set a 20

The

The extent to which a stable water isotope record has been subject to
diffusion can be quantified with the diffusion length

By taking the square of the Fourier transform of
Eq. (

To verify the assumption that the initial signal is independent of
frequency, we have stacked precipitation data from the Global Network
of Isotope in Precipitation (GNIP) database

The power spectral densities of two stacked precipitation records retrieved from the GNIP
database. In light colours the Debrecen data are given; dark colours refer to data from Vernadsky.
The figure on the left shows the PSD as a function of frequency, whereas the figure on the right shows
the natural logarithm of the PSD as a function of the wave number squared (bottom

The PSD of the Debrecen record shows an annual peak at
1.7

The PSDs of the isotope data can be calculated using the
maximum-entropy method (MEM;

Figure a shows the power spectral densities of the

A second parameter that needs to be chosen in the PSD method is the
cut-off frequency (

The choice of the cut-off frequency also has a significant influence
on the resulting value for

In order to find the optimum values for these two parameters, the PSD
method was applied to the 4000 synthetic data sets within each Monte
Carlo run. As the synthetic data sets within an MC run all refer to
the same climatological and glaciological conditions, this approach
will also give insight into how large the stochastic variability is in
the method. In Fig.

Contour plots showing the average deviation from the theoretical value (left) and the
SD of

From these figures it becomes clear that for most solutions the
accuracy (the deviation from the theoretical value) is much higher
than the precision (the SD). The large SD shows how much the obtained
value for

The optimum parameters and the resulting mean values, deviations from theoretical values
and SDs for the differential diffusion lengths of the Monte Carlo runs obtained with the PSD method for
5

In this section a different method for estimating the differential
diffusion length from the stable water isotope records is
presented. The main goal is to come up with a method that is not
sensitive to parameter choices and therefore gives a more robust
estimate of

Illustration of the correlation method for ideal conditions in which the correlation
coefficient of the isotope data before diffusion is 1. The

In reality the initial d-excess signal will not be constant, and
therefore the initial correlation will be below 1. However, for
more realistic initial signals a decrease in correlation as a result
of firn diffusion is also to be expected. To study the correlation method
for more realistic conditions, it was applied to the synthetic data
sets, which have varying d-excess with both a seasonal component and
a random component. For all these data sets the maximum in correlation
coefficient is found as a function of the additional diffusion length
in the

For the data sets discussed above the measurement uncertainty has not
been included in the synthetic data yet. The effect of adding measurement
noise is shown in Fig.

The mean and SD of the differential diffusion length of the 4000 data sets of Monte Carlo run
I as a function of the uncertainty in the isotopic measurement are depicted in the top figures. The
uncertainty is given as the SD of a Gaussian distribution centred around zero from which random numbers
are drawn. The figure on the bottom shows the absolute measurement deviation (as described in the main
text) for two different sampling resolutions. The data points given here are obtained from all six Monte
Carlo runs and for several assigned values for the uncertainty in the

To study this effect in detail, we arbitrarily selected 20 data sets
from each Monte Carlo run. For these data sets the differential
diffusion length is first found for the records without measurement
uncertainty, indicated by

The mean values and deviation from theory and SD for the differential diffusion lengths of the Monte Carlo
runs obtained with the correlation method for 5

In the following section we will compare the performance of the two
methods discussed above in terms of reliably reconstructing the
differential diffusion length. To assess the accuracy of each method, we
compare the mean value for

In general the total error in each of these MC runs is dominated by the SD as this value is significantly larger than the deviation from the theoretical value. However, we will discuss both accuracy and precision separately because the importance of each quantity might vary with application. For example, in quantifying the temperature difference between the Holocene and LGM the accuracy becomes more important. In contrast, in situations where temperature changes are small (for example within the Holocene or around the LGM), the precision is the crucial factor. As will be shown below, the precision can also be influenced by the sampling strategy or replicate analysis, whereas the accuracy is not affected by this.

In terms of accuracy, the PSD method performs slightly better than the
correlation method with a difference between the mean and theoretical
value of less than 2.5 % for all runs, i.e. a systematic error in
the reconstructed temperature of less than
0.3

The numbers given in the discussion above are based on analysis of
20

Sensitivity of the two methods to sample size

From the analysis of synthetic data it follows that the influence of
measurement uncertainty on the precision of the two methods is small
compared to the influence of section length and sampling
resolution. For the PSD method, measurement uncertainty causes
a baseline in the frequency spectra, which needs to be subtracted
before the ratio of the two spectra is calculated. But as the
influence of this subtraction is very small in the low frequency part
of the spectrum, it has negligible influence as long as the cut-off
frequency is chosen correctly. In the correlation method the
measurement uncertainty initially has a significant influence on the
resulting differential diffusion length as is evident from
Fig.

To convert the obtained differential diffusion length to firn
temperature, we use a model similar to those described in

A very sensitive parameter in the calculation of the diffusion length
is the density of the firn as this determines the volume of the pore
space through which most of the diffusion takes place. In the
calculation of

Figure

Density–depth profiles for the B32 ice core drilled close to the main EDML ice core

A 20

Figure

Contour lines of the squared differential diffusion length (

In Sect.

A first requirement for both methods to be applied is that both
isotope records (

After applying this gap-filling procedure, the two methods for
determining the differential diffusion length can be applied. One of
the first steps in the PSD method is to determine the noise level that
needs to be subtracted from the PSD. The determination of the base
line level from the PSD spectrum is, however, very sensitive to
outliers in the isotope record. In Fig.

Part of the

In our EDML record consisting of 3320 samples, we identified 11
samples as outliers in the

Instead of using the noise levels obtained from the PSD, it is also
possible to use a fixed value for the measurement uncertainty based on
repeated measurements of standards. This may lead to more robust
results and is necessary in case the signal at higher frequencies is
of the same or higher magnitude as the noise level. This will be the
case for short diffusion lengths or relatively low resolution
measurements. In Fig.

The squared differential diffusion lengths obtained with the PSD and correlation method

Finally, local surface temperatures for the EDML record are
reconstructed by combining the values of

Local surface temperature reconstructions based on the differential diffusion lengths given in
Fig.

For comparison we also show the temperature records that would be
obtained if the PSD method were applied to single-isotope records.
Although we obtain on average higher temperatures with both

In this paper two methods for retrieving the differential diffusion
length of the stable water isotope signal in ice cores were
investigated. We have shown that temperature reconstruction using the
PSD method on a single isotope can lead to significant biases because
the PSD of the initial signal in precipitation may influence the
obtained diffusion length. Therefore, unless whiteness of the initial
spectrum can be proven, the uncertainty in the reconstructed
temperature is larger for the single-isotope method than for the
double-isotope method. Furthermore,
before the PSD method is applied to ice core data, it needs to be
calibrated by synthetic data. This will allow for an objective
determination of the order of autoregression that is used to compute
the PSD and the cut-off frequency. The quality of such a calibration
depends on how well the synthetic data approximates the real data and
thus on the deposition regime and the climatic conditions
assumed. For the correlation method a calibration is not required as
it is not dependent on the choice of parameters, making this a more
robust method. However, also for this method the use of synthetic data
is required to be able to constrain the uncertainty with which the
differential diffusion length can be obtained. Doing such an analysis
before ice core samples are cut is also useful in determining the best
sampling strategy (length of the section and sampling resolution) to
obtain the required accuracy and precision. Application to part of the
EDML record from the Holocene period has shown that high data quality
is required for diffusion thermometry. Identification of outliers in
the isotope records is possible from the baseline of the PSD
spectra. When the existing data for the EDML ice core (20

The research leading to these results has received funding from the European Research Council under the European Union's Seventh Framework Programme (FP/2007-2013)/ERC Advanced Grant Agreement no. 226172 (MATRICs) awarded to H. Fischer.

This work is also a contribution to the European Project for Ice Coring in Antarctica (EPICA), a joint European Science Foundation–European Commission (EC) scientific programme, funded by the EC and by national contributions from Belgium, Denmark, France, Germany, Italy, the Netherlands, Norway, Sweden, Switzerland and the UK. The main logistic support was provided by IPEV and PNRA (at Dome C) and AWI (at Dronning Maud Land). This is EPICA publication no. 300. Edited by: M. Schneebeli